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External/internal static and kinematic indeterminacy of frame

Short answer

External static indeterminacy is 3, internal is 5 (total 8). Kinematic indeterminacy is 22 with extensible members, or 12 if axial deformation is neglected.

Determine the external and internal degrees of static indeterminacy of the structure shown in figure below. Also determine the kinematic indeterminacy. [Figure: two-storey frame with an internal hinge at top-left; left base fixed, middle base hinged, right top end on a roller.]

Reading of the figure: two-storey, two-bay frame with a hinge at the top-left joint, fixed left base, hinged middle base, and a roller at the right-hand support.

j=9, m=10, r=3+2+1=6, c=1j = 9,\ m = 10,\ r = 3 + 2 + 1 = 6,\ c = 1

Static indeterminacy

  • External: r−3=6−3=3r - 3 = 6 - 3 = 3
  • Internal: 3×(closed loops)−c=3×2−1=53 \times (\text{closed loops}) - c = 3 \times 2 - 1 = 5 (the two upper panels are closed loops; the lower panels are open at the base)
  • Total: Ds=3+5=8D_s = 3 + 5 = 8. Check: 3m+r−3j−c=30+6−27−1=83m + r - 3j - c = 30 + 6 - 27 - 1 = 8.

Kinematic indeterminacy

  • With axial deformation: Dk=3j−r+1=27−6+1=22D_k = 3j - r + 1 = 27 - 6 + 1 = 22
  • Without axial deformation: Dk=22−10=12D_k = 22 - 10 = 12

Answer: external 3, internal 5 (total 8); kinematic 22 (12 if inextensible).

✓ Answer checked by Claude Opus 5.5 (high effort)

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